 Hi, I'm Zor. Welcome to Indizora Education. I would like to do some calculations related to launching satellites today. Very, very simple, very basic calculations. We are talking only about the speed of the satellite, which is rotating around the planet. And I need that speed, basically, based on certain parameters. So, we have a few laws, basically two laws, and both are related to Sir Isaac Newton. Okay, one law is the gravitational law. If you have two objects, now we are talking about ideal situations when both objects are either point masses or spherical, with the center of the gravity in the center of the sphere. Now, Newton's gravitational law sends the following. Hope is familiar to you. So, this is the gravitational constant, universal constant. This is the mass of the planet. This is the mass of satellite. And this is the radius from a center to a center, center of the planet to center of the satellite. The radius of orbit, we want that satellite to be rotating around the planet. So, it includes, actually, the size of the planet. So, it's from a center of the sphere of the planet to a center of the satellite. So, we know this. Great. This is the force which acts on the satellite when it's rotating. Now, if that would be the only force, then the satellite would just fall on the planet. Now, we have to launch it so it goes around the planet with a certain speed. And considering this is a circular orbit, there is a centripetal acceleration. And if there is a centripetal acceleration, it must be centripetal force, which is basically equal to this one. So, these are two different kind of tendencies which are acting on the same object. And it's just two different ways of calculating this. Because if you would like to, if this is the planet and this is the orbit, if there is no centripetal force, the satellite would go along the tangential line. So, there might be a force which pulls it in. And that's a centripetal force. And since this is actually the source of this force, we are equalizing centripetal force which can be calculated as m times a where a is centripetal acceleration. And this one, these are the same forces just calculated from two different viewpoints. This is from the viewpoint of the gravitation law and this is from the viewpoint of pure movement laws. And this is kinematics, if you wish, because acceleration is a kinematic characteristic. Okay, so what is A? We are talking about the speed, linear speed, the satellite is supposed to be circulating the planet. So, how the speed is related to acceleration. So, we are talking about linear speed. We did calculate it a few times, but in any case. Basically, let me just write it down. A is equal to b squared divided by r. Now, how to obtain this formula is very easy. If you have a circular motion, the best and probably easier way is to describe x and y characteristics of this. It's r cosine omega t and this is r sine omega t. Now, the speed is the first derivative, right? So, the first derivative is, I'm using something from the calculus, but basically that's something which people did many, many times before. The first derivative is what from cosine is minus r omega sine omega t from y is cosine, so it's r internal cosine omega t. The second derivative which gives me the acceleration, it's minus r omega square cosine omega t and y is equal to minus r omega square sine omega t. Acceleration is equal to the length of this vector, vector of acceleration by coordinates which gives me r omega square and linear speed is equal to radius times angular speed, right? So, from this, acceleration is equal to r omega square omega is equal to v divided by r, so it's v square divided by r square and we have formula as I said. Now, it's a very, very quick derivation in any case that's supposed to be addressed in kinematics of rotation which we already covered it before. So, this is equal to m v square divided by r, okay? Now, this is the same force, so I'm just equalizing that and that's where we are getting the speed. Now, what's absolutely amazing, at least for me, is this. So, it doesn't depend on the mass of the satellite, it depends only on the radius. v square is equal to g times m divided by r and v is equal to square root of gm divided by r. So, g is gravitational constant, m is the planet mass and r is the radius distance between the center of the planet and the center of satellite. So, this is the speed which we have to really give to a satellite like a push, so whenever the engine of a rocket brings the satellite to a proper distance from the planet, then it should turn and push it with this speed. As soon as the speed is sufficient, you can turn off the engine. Now, this is a very complicated process because first you have to really bring the satellite up. Now, that actually depends on the mass. I mean, the heavier the satellite, the more difficult it is to bring it up to the orbit. The engine must be much stronger. And then you have to really turn it in such a way that it retains the distance needed for you and then goes. Okay, what else can be derived from this? Well, actually we can derive the period. Now, period is the time it circulates around the planet. So, what is the period? If we have the speed, now the distance we also have, the distance is 2 pi r, right? So, we divide 2 pi r, t is equal by speed, and what do we have? We have 2 pi r, and t goes another way around, r divided by gm. Or we can put r inside the square root. It would be square root of r to the third degree divided by gm. Okay, now gm has nothing to do with general motors. It's gravitational constant in the mass. The greater the mass with the same radius, the faster it should actually rotate, obviously, because the mass is pulling the satellite stronger if it has a bigger mass. So, that's basically it. That's all I wanted to say about the speed and the period of a satellite which is freely without any engine working. Freely rotating around the Earth. It's supposed to be somehow calculated if you want to launch a satellite on a specific orbit. Okay, I suggest you to read the notes for this lecture, which is presented on Unisor.com. The course is called Physics for Chains, and it's mechanics about gravitation. You will find this lecture. Thank you very much and good luck.